I was wondering, why and when is it important to fund your contract with ether? Is this really only ever if you want to send ether from your contract?
Does a contract ever need to pay gas for something rather than the caller?
NO, till yet, contract cannot pay the gas for the execution of a transaction. If the function of a contract, in turn, call the function of another contract, gas will be provided by the user calling the function of first contract.
This can add a bit more : Can contracts pay the gas instead of the message sender?